Arlinskii's iteration and its applications

TL;DR

This paper introduces a unified, elementary approach to Lebesgue-type decompositions across various mathematical objects using Arlinskii's iteration, highlighting the relation with the parallel sum operation.

Contribution

It generalizes Arlinskii's method to establish Lebesgue-type decompositions for nonnegative sesquilinear forms and applies it to related structures with a unified proof technique.

Findings

Proved existence of Lebesgue-type decomposition for sesquilinear forms

Derived analogous results for measures and operator functions

Unified elementary proof method for multiple decomposition theorems

Abstract

Several Lebesgue-type decomposition theorems in analysis have a strong relation to the operation called: parallel sum. The aim of this paper is to investigate this relation from a new point of view. Namely, using a natural generalization of Arlinskii's approach (which identifies the singular part as a fixed point of a single-variable map) we prove the existence of a Lebesgue-type decomposition for nonnegative sesquilinear forms. As applications, we also show that how this approach can be used to derive analogous results for representable functionals, nonnegative finitely additive measures, and positive definite operator functions. The focus is on the fact that each theorem can be proved with the same completely elementary method.